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G = C24.3Q8order 128 = 27

2nd non-split extension by C24 of Q8 acting via Q8/C2=C22

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C24.3Q8, C22⋊C8.8C4, (C2×C4).10C42, C4.34(C23⋊C4), C23.43(C4⋊C4), (C22×C4).641D4, (C2×M4(2)).1C4, C24.4C4.7C2, C2.6(C4.C42), C22.6(C8.C4), (C23×C4).194C22, C2.8(C23.9D4), C22.48(C2.C42), (C2×C22⋊C8).6C2, (C22×C4).153(C2×C4), (C2×C4).301(C22⋊C4), SmallGroup(128,30)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.3Q8
C1C2C22C2×C4C22×C4C23×C4C2×C22⋊C8 — C24.3Q8
C1C22C2×C4 — C24.3Q8
C1C2×C4C23×C4 — C24.3Q8
C1C2C22C23×C4 — C24.3Q8

Generators and relations for C24.3Q8
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=c, f2=bce2, ab=ba, ac=ca, eae-1=faf-1=ad=da, bc=cb, bd=db, be=eb, fbf-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=acde3 >

Subgroups: 200 in 95 conjugacy classes, 36 normal (12 characteristic)
C1, C2, C2 [×2], C2 [×5], C4 [×2], C4 [×3], C22, C22 [×4], C22 [×11], C8 [×6], C2×C4 [×2], C2×C4 [×2], C2×C4 [×9], C23, C23 [×2], C23 [×5], C2×C8 [×10], M4(2) [×2], C22×C4 [×2], C22×C4 [×4], C22×C4 [×2], C24, C22⋊C8 [×4], C22⋊C8 [×4], C22×C8 [×2], C2×M4(2) [×2], C23×C4, C2×C22⋊C8 [×2], C24.4C4, C24.3Q8
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2.C42, C23⋊C4 [×2], C8.C4 [×4], C4.C42 [×2], C23.9D4, C24.3Q8

Smallest permutation representation of C24.3Q8
On 32 points
Generators in S32
(2 18)(4 20)(6 22)(8 24)(10 28)(12 30)(14 32)(16 26)
(1 5)(2 6)(3 7)(4 8)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 25)(16 26)(17 21)(18 22)(19 23)(20 24)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 25)(16 26)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)
(1 14 3 30 5 10 7 26)(2 13 4 29 6 9 8 25)(11 22 27 24 15 18 31 20)(12 21 28 23 16 17 32 19)

G:=sub<Sym(32)| (2,18)(4,20)(6,22)(8,24)(10,28)(12,30)(14,32)(16,26), (1,5)(2,6)(3,7)(4,8)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,25)(16,26)(17,21)(18,22)(19,23)(20,24), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,25)(16,26), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32), (1,14,3,30,5,10,7,26)(2,13,4,29,6,9,8,25)(11,22,27,24,15,18,31,20)(12,21,28,23,16,17,32,19)>;

G:=Group( (2,18)(4,20)(6,22)(8,24)(10,28)(12,30)(14,32)(16,26), (1,5)(2,6)(3,7)(4,8)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,25)(16,26)(17,21)(18,22)(19,23)(20,24), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,25)(16,26), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32), (1,14,3,30,5,10,7,26)(2,13,4,29,6,9,8,25)(11,22,27,24,15,18,31,20)(12,21,28,23,16,17,32,19) );

G=PermutationGroup([(2,18),(4,20),(6,22),(8,24),(10,28),(12,30),(14,32),(16,26)], [(1,5),(2,6),(3,7),(4,8),(9,27),(10,28),(11,29),(12,30),(13,31),(14,32),(15,25),(16,26),(17,21),(18,22),(19,23),(20,24)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32)], [(1,17),(2,18),(3,19),(4,20),(5,21),(6,22),(7,23),(8,24),(9,27),(10,28),(11,29),(12,30),(13,31),(14,32),(15,25),(16,26)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32)], [(1,14,3,30,5,10,7,26),(2,13,4,29,6,9,8,25),(11,22,27,24,15,18,31,20),(12,21,28,23,16,17,32,19)])

38 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I8A···8P8Q8R8S8T
order1222222224444444448···88888
size1111222241111222244···48888

38 irreducible representations

dim111112224
type++++-+
imageC1C2C2C4C4D4Q8C8.C4C23⋊C4
kernelC24.3Q8C2×C22⋊C8C24.4C4C22⋊C8C2×M4(2)C22×C4C24C22C4
# reps1218431162

Matrix representation of C24.3Q8 in GL4(𝔽17) generated by

1000
0100
0010
00916
,
16000
1100
00160
00016
,
16000
01600
00160
00016
,
1000
0100
00160
00016
,
9000
5200
00915
00138
,
161500
11100
0041
00513
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,1,9,0,0,0,16],[16,1,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[9,5,0,0,0,2,0,0,0,0,9,13,0,0,15,8],[16,11,0,0,15,1,0,0,0,0,4,5,0,0,1,13] >;

C24.3Q8 in GAP, Magma, Sage, TeX

C_2^4._3Q_8
% in TeX

G:=Group("C2^4.3Q8");
// GroupNames label

G:=SmallGroup(128,30);
// by ID

G=gap.SmallGroup(128,30);
# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,723,184,136,3924,102]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=c,f^2=b*c*e^2,a*b=b*a,a*c=c*a,e*a*e^-1=f*a*f^-1=a*d=d*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f^-1=b*c*d,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=a*c*d*e^3>;
// generators/relations

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